Equation of Confocal Conics/Formulation 2
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Definition
The equation:
- $(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
where:
- $\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
- $c$ is a (strictly) positive constant
- $a$ is a (strictly) positive parameter
defines the set of all confocal conics whose foci are at $\tuple {\pm c, 0}$.
Proof
Let $a > c$.
Then from Equation of Confocal Ellipses: Formulation 2, $(1)$ defines the set of all confocal ellipses whose foci are at $\tuple {\pm c, 0}$.
Let $a < c$.
Then from Equation of Confocal Hyperbolas, $(1)$ defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm c, 0}$.
Hence the result.
$\blacksquare$
Also see
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $6$